On the Universal Tiling Conjecture in Dimension One
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On the Universal Tiling Conjecture in Dimension One Dorin Ervin Dutkay · Palle E.T. Jorgensen
Received: 4 September 2012 / Revised: 24 January 2013 / Published online: 13 March 2013 © Springer Science+Business Media New York 2013
Abstract We show that the spectral-tile implication in the Fuglede conjecture in dimension 1 is equivalent to a Universal Tiling Conjecture and also to similar forms of the same implication for some simpler sets, such as unions of intervals with rational or integer endpoints. Keywords Fuglede conjecture · Spectrum · Tiling · Spectral pairs · Fourier analysis Mathematics Subject Classification (2010) 42A32 · 05B45
1 Introduction Definition 1.1 For λ ∈ R we denote by eλ (x) := e2πiλx ,
(x ∈ R)
Let Ω be Lebesgue measurable subset of R with finite Lebesgue measure. We say that Ω is spectral if there exists a subset Λ of R such that {eλ : λ ∈ Λ} is an orthogonal basis for L2 (Ω). In this case Λ is called a spectrum for Ω. We say that Ω tiles R by translations if there exists a subset T of R such that {Ω + t : t ∈ T } is a partition of R, up to Lebesgue measure zero. Communicated by Karlheinz Gröchenig. D.E. Dutkay () Department of Mathematics, University of Central Florida, 4000 Central Florida Blvd., P.O. Box 161364, Orlando, FL 32816-1364, USA e-mail: [email protected] P.E.T. Jorgensen Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA 52242-1419, USA e-mail: [email protected]
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J Fourier Anal Appl (2013) 19:467–477
Fuglede’s conjecture was stated for arbitrary finite dimension in [3]. It asserts that the tiling and the spectral properties are equivalent. Tao [11] disproved one direction in the Fuglede conjecture in dimensions 5 or higher: there exists a union of cubes which is spectral but does not tile. Later, Tao’s counterexample was improved to disprove both directions in Fuglede’s conjecture for dimensions 3 or higher [2, 5]. In the cases of dimensions 1 and 2, both directions are still open. We state here Fuglede’s conjecture in dimension 1: Conjecture 1.2 [3] A subset Ω of R of finite Lebesgue measure is spectral if and only if it tiles R by translations. We focus on the spectral-tile implication in the Fuglede conjecture and present some equivalent statements. One of the main ingredients that we will use is the fact that any spectrum is periodic (see [1, 4]). There are good reasons for our focus on the cases of the conjectures in one dimension. One reason is periodicity (see the next definition): it is known, in 1D, that the possible sets Λ serving as candidates for spectra, in the sense of Fuglede’s conjecture (Conjecture 1.2), must be periodic. A second reason lies in the difference, from 1D to 2D, in the possibilities for geometric configurations of translation sets. The Universal Tiling Conjecture (Conjecture 1.4) suggests a reduction of the implication from spectrum to tile in Fuglede’s conjecture, to a consideration of finite subsets of Z. Hence computations for the problems in 1D are arithmetic in nature, as opposed to geometric; and
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